Editing Riesz representation theorem (section)

===Statement===

{{Math theorem
|name = {{visible anchor|Riesz representation theorem}}
|math_statement =
Let <math>H</math> be a [[Hilbert space]] whose [[inner product]] <math>\left\langle x, y \right\rangle</math> is linear in its {{em|first}} argument and [[Antilinear map|antilinear]] in its second argument and let <math>\langle y \mid x \rangle := \langle x, y \rangle</math> be the corresponding physics notation. For every continuous linear functional <math>\varphi \in H^*,</math> there exists a unique vector <math>f_{\varphi} \in H,</math> called the {{em|{{visible anchor|Riesz representation|Riesz representative}} of <math>\varphi,</math>}}<!-- the term "Riesz representation of" seems to be more commonly used than "Riesz representative of" --> such that<ref>{{harvnb|Roman|2008|loc=p. 351 Theorem 13.32}}</ref>
<math display=block>\varphi(x) = \left\langle x, f_{\varphi} \right\rangle = \left\langle f_\varphi \mid x \right\rangle \quad \text{ for all } x \in H.</math>

Importantly for {{em|complex}} Hilbert spaces, <math>f_{\varphi}</math> is always located in the {{em|antilinear}} coordinate of the inner product.<ref name="ImportanceOfLocationOfRieszRep" group="note" />

Furthermore, the length of the representation vector is equal to the norm of the functional:
<math display=block>\left\|f_\varphi\right\|_H = \|\varphi\|_{H^*},</math> and <math>f_{\varphi}</math> is the unique vector <math>f_{\varphi} \in \left(\ker \varphi\right)^{\bot}</math> with <math>\varphi\left(f_{\varphi}\right) = \|\varphi\|^2.</math> 
It is also the unique element of minimum norm in <math>C := \varphi^{-1}\left(\|\varphi\|^2\right)</math>; that is to say, <math>f_{\varphi}</math> is the unique element of <math>C</math> satisfying <math>\left\|f_{\varphi}\right\| = \inf_{c \in C} \|c\|.</math>
Moreover, any non-zero <math>q \in (\ker \varphi)^{\bot}</math> can be written as <math>q = \left(\|q\|^2 /\, \overline{\varphi(q)}\right)\  f_{\varphi}.</math>
}}

{{Math theorem
| name = Corollary
| math_statement = The {{em|canonical map from <math>H</math> into its dual}} <math>H^*</math>{{sfn|Trèves|2006|pp=112–123}} is the [[Injective map|injective]] [[Antilinear map|{{em|anti}}linear operator]] [[isometry]]<ref group=note name="AntilinearIsometryDef" />{{sfn|Trèves|2006|pp=112–123}} 
<math display=block>\begin{alignat}{4}
\Phi :\;&& H &&\;\to    \;& H^* \\[0.3ex]
        && y &&\;\mapsto\;& \langle \,\cdot\,, y \rangle = \langle y | \,\cdot\, \rangle \\
\end{alignat}</math>
The Riesz representation theorem states that this map is [[Surjective map|surjective]] (and thus [[Bijective map|bijective]]) when <math>H</math> is complete and that its inverse is the [[Bijective map|bijective]] [[Isometry|isometric]] antilinear isomorphism
<math display=block>\begin{alignat}{4}
\Phi^{-1} :\;&& H^*     &&\;\to    \;& H \\[0.3ex]
             && \varphi &&\;\mapsto\;& f_{\varphi} \\
\end{alignat}.</math>
Consequently, {{em|every}} continuous linear functional on the Hilbert space <math>H</math> can be written uniquely in the form <math>\langle y\, | \,\cdot\, \rangle</math>{{sfn|Trèves|2006|pp=112–123}} where <math>\|\langle y\,| \cdot \rangle\|_{H^*} = \|y\|_H</math> for every <math>y \in H.</math> 
The assignment <math>y \mapsto \langle y, \cdot \rangle = \langle \cdot\,| \,y \rangle</math> can also be viewed as a bijective {{em|linear}} isometry <math>H \to \overline{H}^*</math> into the [[Antilinear map|anti-dual space]] of <math>H,</math>{{sfn|Trèves|2006|pp=112–123}} which is the [[complex conjugate vector space]] of the [[continuous dual space]] <math>H^*.</math> 

The inner products on <math>H</math> and <math>H^*</math> are related by 
<math display=block>\left\langle \Phi h, \Phi k \right\rangle_{H^*} = \overline{\langle h, k \rangle}_H = \langle k, h \rangle_H \quad \text{ for all } h, k \in H</math>
and similarly, 
<math display=block>\left\langle \Phi^{-1} \varphi, \Phi^{-1} \psi \right\rangle_H = \overline{\langle \varphi, \psi \rangle}_{H^*} = \left\langle \psi, \varphi \right\rangle_{H^*} \quad \text{ for all } \varphi, \psi \in H^*.</math>

The set <math>C := \varphi^{-1}\left(\|\varphi\|^2\right)</math> satisfies <math>C = f_{\varphi} + \ker \varphi</math> and <math>C - f_{\varphi} = \ker \varphi</math> so when <math>f_{\varphi} \neq 0</math> then <math>C</math> can be interpreted as being the [[affine hyperplane]]<ref group=note name="VectorSpaceStructureOnAffineHyperplanesInducedByDualSpace" /> that is parallel to the vector subspace <math>\ker \varphi</math> and contains <math>f_{\varphi}.</math> 

For <math>y \in H,</math> the physics notation for the functional <math>\Phi(y) \in H^*</math> is the bra <math>\langle y |,</math> where explicitly this means that <math>\langle y | := \Phi(y),</math> which complements the ket notation <math>| y \rangle</math> defined by <math>| y \rangle := y.</math> 
In the mathematical treatment of [[quantum mechanics]], the theorem can be seen as a justification for the popular [[bra–ket notation]]. The theorem says that, every bra <math>\langle\psi\,|</math> has a corresponding ket <math>|\,\psi\rangle,</math> and the latter is unique.
}}

Historically, the theorem is often attributed simultaneously to [[Frigyes Riesz|Riesz]] and [[Maurice René Fréchet|Fréchet]] in 1907 (see references). 

{{collapse top|title=Proof{{sfn|Rudin|1991|pp=307−309}}|left=true}}
Let <math>\mathbb{F}</math> denote the underlying scalar field of <math>H.</math> 

{{em|Proof of norm formula:}} 

Fix <math>y \in H.</math>  
Define <math>\Lambda : H \to \mathbb{F}</math> by <math>\Lambda(z) := \langle \,y\, | \,z\, \rangle,</math> which is a linear functional on <math>H</math> since <math>z</math> is in the linear argument. 
By the [[Cauchy–Schwarz inequality]], 
<math display=block>|\Lambda(z)| = |\langle \,y\, | \,z\, \rangle| \leq \|y\| \|z\|</math>
which shows that <math>\Lambda</math> is bounded (equivalently, [[Continuous linear functional|continuous]]) and that <math>\|\Lambda\| \leq \|y\|.</math> 
It remains to show that <math>\|y\| \leq \|\Lambda\|.</math> 
By using <math>y</math> in place of <math>z,</math> it follows that 
<math display=block>\|y\|^2 = \langle \,y\, | \,y\, \rangle = \Lambda y = |\Lambda(y)| \leq \|\Lambda\| \|y\|</math>
(the equality <math>\Lambda y = |\Lambda(y)|</math> holds because <math>\Lambda y = \|y\|^2 \geq 0</math> is real and non-negative). 
Thus that <math>\|\Lambda\| = \|y\|.</math> 
<math>\blacksquare</math>

The proof above did not use the fact that <math>H</math> is [[Complete metric space|complete]], which shows that the formula for the norm <math>\|\langle \,y\, | \,\cdot\, \rangle\|_{H^*} = \|y\|_H</math> holds more generally for all [[inner product space]]s. 

{{hr|1}}

{{em|Proof that a Riesz representation of <math>\varphi</math> is unique:}} 

Suppose <math>f, g \in H</math> are such that <math>\varphi(z) = \langle \,f\, | \,z\, \rangle</math> and <math>\varphi(z) = \langle \,g\, | \,z\, \rangle</math> for all <math>z \in H.</math> 
Then 
<math display=block>\langle \,f - g\, | \,z\, \rangle = \langle \,f\, | \,z\, \rangle - \langle \,g\, | \,z\, \rangle = \varphi(z) - \varphi(z) = 0 \quad \text{ for all } z \in H</math>
which shows that <math>\Lambda := \langle \,f - g\, | \,\cdot\, \rangle</math> is the constant <math>0</math> linear functional. 
Consequently <math>0 = \|\langle \,f - g\, | \,\cdot\, \rangle\| = \|f - g\|,</math> which implies that <math>f - g = 0.</math> 
<math>\blacksquare</math>

{{hr|1}}

{{em|Proof that a vector <math>f_{\varphi}</math> representing <math>\varphi</math> exists:}} 

Let <math>K := \ker \varphi := \{ m \in H : \varphi(m) = 0 \}.</math> 
If <math>K = H</math> (or equivalently, if <math>\varphi = 0</math>) then taking <math>f_{\varphi} := 0</math> completes the proof so assume that <math>K \neq H</math> and <math>\varphi \neq 0.</math> 
The continuity of <math>\varphi</math> implies that <math>K</math> is a closed subspace of <math>H</math> (because <math>K = \varphi^{-1}(\{ 0 \})</math> and <math>\{ 0 \}</math> is a closed subset of <math>\mathbb{F}</math>). 
Let 
<math display=block>K^{\bot} := \{ v \in H ~:~ \langle \,v\, | \,k\, \rangle = 0 ~ \text{ for all } k \in K\}</math> 
denote the [[orthogonal complement]] of <math>K</math> in <math>H.</math>
Because <math>K</math> is closed and <math>H</math> is a Hilbert space,<ref group=note>Showing that there is a non-zero vector <math>v</math> in <math>K^{\bot}</math> relies on the continuity of <math>\phi</math> and the [[Cauchy completeness]] of <math>H.</math> This is the only place in the proof in which these properties are used.</ref> <math>H</math> can be written as the direct sum <math>H = K \oplus K^{\bot}</math><ref group=note>Technically, <math>H = K \oplus K^{\bot}</math> means that the addition map <math>K \times K^{\bot} \to H</math> defined by <math>(k, p) \mapsto k + p</math> is a surjective [[linear isomorphism]] and [[homeomorphism]]. See the article on [[complemented subspace]]s for more details.</ref> (a proof of this is given in the article on the [[Hilbert projection theorem]]). 
Because <math>K \neq H,</math> there exists some non-zero <math>p \in K^{\bot}.</math> 
For any <math>h \in H,</math>
<math display=block>\varphi[(\varphi h) p - (\varphi p) h] 
~=~ \varphi[(\varphi h) p] - \varphi[(\varphi p) h] 
~=~ (\varphi h) \varphi p - (\varphi p) \varphi h 
= 0,</math>
which shows that <math>(\varphi h) p - (\varphi p) h ~\in~ \ker \varphi = K,</math> where now <math>p \in K^{\bot}</math> implies 
<math display=block>0 = \langle \,p\, | \,(\varphi h) p - (\varphi p) h\, \rangle 
~=~ \langle \,p\, | \,(\varphi h) p \, \rangle - \langle \,p\, | \,(\varphi p) h\, \rangle 
~=~ (\varphi h) \langle \,p\, | \,p \, \rangle - (\varphi p) \langle \,p\, | \,h\, \rangle.</math> 
Solving for <math>\varphi h</math> shows that 
<math display=block>\varphi h 
= \frac{(\varphi p) \langle \,p\, | \,h\, \rangle}{\|p\|^2} 
= \left\langle \,\frac{\overline{\varphi p}}{\|p\|^2} p\, \Bigg| \,h\, \right\rangle \quad \text{ for every } h \in H,</math>
which proves that the vector <math>f_{\varphi} := \frac{\overline{\varphi p}}{\|p\|^2} p</math> satisfies 
<math>\varphi h = \langle \,f_{\varphi}\, | \,h\, \rangle \text{ for every } h \in H.</math> 

Applying the norm formula that was proved above with <math>y := f_{\varphi}</math> shows that <math>\|\varphi\|_{H^*} = \left\|\left\langle \,f_{\varphi}\, | \,\cdot\, \right\rangle\right\|_{H^*} = \left\|f_{\varphi}\right\|_H.</math> 
Also, the vector <math>u := \frac{p}{\|p\|}</math> has norm <math>\|u\| = 1</math> and satisfies <math>f_{\varphi} := \overline{\varphi(u)} u.</math> 
<math>\blacksquare</math>

{{hr|1}}

It can now be deduced that <math>K^{\bot}</math> is <math>1</math>-dimensional when <math>\varphi \neq 0.</math> 
Let <math>q \in K^{\bot}</math> be any non-zero vector. Replacing <math>p</math> with <math>q</math> in the proof above shows that the vector <math>g := \frac{\overline{\varphi q}}{\|q\|^2} q</math> satisfies <math>\varphi(h) = \langle \,g\, | \,h\, \rangle</math> for every <math>h \in H.</math> The uniqueness of the (non-zero) vector <math>f_{\varphi}</math> representing <math>\varphi</math> implies that <math>f_{\varphi} = g,</math> which in turn implies that <math>\overline{\varphi q} \neq 0</math> and <math>q = \frac{\|q\|^2}{\overline{\varphi q}} f_{\varphi}.</math> Thus every vector in <math>K^{\bot}</math> is a scalar multiple of <math>f_{\varphi}.</math> <math>\blacksquare</math>

The formulas for the inner products follow from the [[polarization identity]]. 

{{collapse bottom}}